Building N-sided boxes and polygonal catch-all bowls introduces compound geometry that goes beyond basic miter cuts. Whether you are making a hexagonal box, an octagonal tray, or a multi-sided bowl with angled walls, the accuracy of your angle calculations directly determines how cleanly the project assembles.

This article explains the math behind those angles and shows how to apply them correctly at the table saw or miter saw.

1. Understanding the Geometry of N-Sided Forms

Any N-sided box or bowl is based on a regular polygon, meaning:

• All sides are equal length

• All interior angles are equal

• The total rotation around the shape is 360°

Each side must rotate a specific amount relative to the next to close the form cleanly.

2. The Key Angle: Exterior Angle

The most important value is the exterior angle, not the interior angle.

Exterior Angle Formula

Exterior Angle = 360° ÷ N

Where N is the number of sides.

Examples

• 6-sided box (hexagon):
  360 ÷ 6 = 60°

• 8-sided box (octagon):
  360 ÷ 8 = 45°

• 12-sided bowl:
  360 ÷ 12 = 30°

This exterior angle represents the total rotation from one side to the next.

3. Miter Angle for Each Side

Each joint splits that rotation evenly between two mating edges.

Miter Angle Formula

Miter Angle = (360° ÷ N) ÷ 2

Examples

• Hexagon:
  (360 ÷ 6) ÷ 2 = 30°

• Octagon:
  (360 ÷ 8) ÷ 2 = 22.5°

• 12-sided bowl:
  (360 ÷ 12) ÷ 2 = 15°

This is the angle you set on your miter gauge or saw when the walls are vertical.

4. Adding Wall Slope (Compound Angles)

Catch-all bowls typically have angled walls, not vertical ones. This introduces a compound angle:

• Miter angle (rotation around the shape)

• Bevel angle (wall lean)

Wall Angle Definition

The wall angle is measured from vertical.

Example:

• Vertical wall = 0°

• Slightly flared bowl = 5°–10°

• Aggressive bowl profile = 12°–18°

5. Calculating the Bevel Angle

The bevel angle is simply the wall angle you want.

Example:

• Desired wall slope = 10°

• Bevel setting = 10°

This tilts the blade to create the angled sidewall.

6. Adjusting the Miter for a Compound Cut

When beveling the blade, the effective miter angle changes slightly. For most woodworking applications, the following formula is accurate enough for tight glue-ups.

Compound Miter Formula

Adjusted Miter =
arctan(
  sin(360° ÷ N) ÷
  (cos(360° ÷ N) + cos(Bevel Angle))
)

In practice:

• For shallow wall angles (≤10°), many builders use the standard miter angle without adjustment.

• For steep bowls or high side counts, calculating or test-cutting becomes important.

7. Practical Rule-of-Thumb (Shop-Friendly)

For most catch-all bowls and boxes:

1. Calculate the standard miter: (360 ÷ N) ÷ 2

2. Set the bevel to the desired wall angle.

3. Cut two test pieces.

4. Dry-fit four to six sides.

5. Fine-tune the miter by ±0.2° if gaps appear.

This approach is faster and more reliable than chasing math alone.

8. Common Pitfalls

• Using interior angles instead of exterior angles

• Rounding too aggressively (tenths of a degree matter)

• Skipping test cuts

• Inconsistent stock thickness

• Blade deflection during bevel cuts

Precision matters more as the number of sides increases.

9. Example: 8-Sided Catch-All Bowl

Target:

• 8 sides

• 12° wall slope

Calculations:

• Exterior angle: 360 ÷ 8 = 45°

• Miter angle: 45 ÷ 2 = 22.5°

• Bevel angle: 12°

Setup:

• Miter gauge: 22.5°

• Blade bevel: 12°

• Test fit before full glue-up

10. Why This Matters

Accurate angle calculations:

• Reduce sanding and filler

• Improve glue-up strength

• Produce cleaner, more professional results

• Enable repeatable designs for small-batch production

Once mastered, N-sided geometry opens the door to advanced boxes, bowls, trays, and architectural forms that stand out immediately.

Final Takeaway

Every N-sided form is governed by the same rule:
360° divided by the number of sides.
From there, precision, testing, and consistency do the rest.

If you can calculate the angles, you can build almost any polygonal form with confidence.

Try the N-Sided Calculator